Regarding the horizontal space, a particular one by imposing orthogonal conditions is proposed in [1]}. This horizontal space has also been exploited in [2]} for the development of Riemannian quotient algorithms. Moreover, given a Riemannian metric \(\bar{g} \left(\cdot ,\cdot \right)\) ,
one can construct the horizontal space \({\cal H}_{\bar{{\cal X}}}\)  which is orthogonal complementary to the vertical space \({\cal V}_{\bar{{\cal X}}}\) :
\({\cal H}_{\bar{{\cal X}}} = \left\lbrace \bar{\xi }\in T_{\bar{{\cal X}}}\overline{{\cal M}}_{\mathbf {r}}: \bar{g}_{\bar{{\cal X}}}\left(\bar{\xi },\bar{\eta }\right)= 0,~\text{for all}~\bar{\eta }\in {\cal V}_{\bar{{\cal X}}}\right\rbrace .\) 
